## Multiplier Hopf algebras

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- by A. Van Daele PDF
- Trans. Amer. Math. Soc.
**342**(1994), 917-932 Request permission

## Abstract:

In this paper we generalize the notion of Hopf algebra. We consider an algebra*A*, with or without identity, and a homomorphism $\Delta$ from

*A*to the multiplier algebra $M(A \otimes A)$ of $A \otimes A$. We impose certain conditions on $\Delta$ (such as coassociativity). Then we call the pair $(A,\Delta )$ a multiplier Hopf algebra. The motivating example is the case where

*A*is the algebra of complex, finitely supported functions on a group

*G*and where $(\Delta f)(s,t) = f(st)$ with $s,t \in G$ and $f \in A$. We prove the existence of a counit and an antipode. If

*A*has an identity, we have a usual Hopf algebra. We also consider the case where

*A*is a $\ast$-algebra. Then we show that (a large enough) subspace of the dual space can also be made into a $\ast$-algebra.

## References

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## Additional Information

- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**342**(1994), 917-932 - MSC: Primary 16W30
- DOI: https://doi.org/10.1090/S0002-9947-1994-1220906-5
- MathSciNet review: 1220906